As we know, a room mode or standing wave occurs when half a wavelength of a sound fits neatly between two opposing boundaries. When the wave reflects it adds with the first wave and creates an even bigger one. This also happens at multiples of the half wavelength. That is, half a cycle, 1 cycle, 1.5 cycles, 2 cycles, etc..
So if we know a room has a length of 5meters apart we can calculate the main length modes to be at: 171 (half speed of sound) / 5 (length in meters) = 34Hz. Next mode is at 68Hz, 102Hz, 137Hz, 172Hz etc..
Axial (Two Boundary) Standing Wave
Image taken from http://www.mcsquared.com/modecalc.htm
These simple two boundary standing waves are called Axial. Standing waves can also form between four boundaries, kind of running around the walls of the room like Neo in The Matrix. These are called Tangential modes.
Tangential Standing Wave
Image taken from http://www.mcsquared.com/modecalc.htm
Finally, standing waves can also form around 6 boundaries and are called Oblique room modes. Each type looses more energy as it requires more reflections to form and so the level is lower.
Oblique Standing Wave
Image taken from http://www.santafevisions.com/csf/html/lectures/022_environment_III.htm
The formula to find a rooms modes, Axial, Tangential and Oblique is:
You might have guessed that L, W and H are the rooms Length, Width and Height.
P, Q and R denote whether the mode is Axial, Tangential or Oblique. If P=1, Q=0 and R = 0, then the standing wave is only in the 'Length' dimension and is an Axial mode. 1,1,0 would incorporate 'L' and 'W' dimensions and is a Tangential standing wave.
You get the idea. I won't bore you with making all the calculations since these days we have computers! Actually, if you read on, I'll explain why a simple mode calculator isn't much use to getting the best from a listening room, unless you want to find the best ratios to build a room from scratch.
If you are building a listening room from the ground up, consider that we want a room where it's natural resonances are as evenly spaced as possible. That way, with a bit of damping, they can be smoothed out to give a nice response without peaks or dips. Lists of such ratios exist, such as those from Sepmeyer or Louden, but I'm not going to put them here. If you are building a new listening room and you don't already know the best sizes or where to find them, hire an acoustics expert or you could be wasting a great opportunity to get the best sound you ever heard.
So far we have looked at a rooms natural resonant frequencies. However, to energise all axial, tangential and oblique modes the speaker would need to be mounted in one of the rooms corners where length, width and height boundaries meet (tri-corners) and thus all modes also meet. The effect of placing the sound source some place other than the boundary is that some of the room modes will be less easily driven. This is because the speaker could be trying to drive the mode from a null or node, a point where the direct and reflected wave cancel each other. This is in contrast to an anti-node where the waves combine to create maximum pressure fluctuations and the standing wave behaviour is very easily excited.
The following link is a calculator that works out the rooms natural modes, and then allows you to see the response that occurs when we move the listening position and the sound source.
There are a three of factors the above calculator ignores, though. These are comb-filtering from boundary reflections, boundary coupling, and the effect of multiple sound sources interfering with each other. I will discuss these issues below.
As we know, sound can reflect without causing a standing wave. In this case it is possible that the reflected wave cancels or combines with the original source to varying degrees. This depends on the distance of the sound source from the boundary, and the wavelength of the sound. The effect is called comb-filtering since the dips and peaks occur at steady increments, however as frequency rises we tend to see more energy lost in reflection and so the severity is lessened.
The frequency response changes occur throughout the room, since the reflection interferes at the source of the sound. This is in contrast to standing waves which have hot and cold spots throughout the room. The graph below shows the result of a speaker being placed 1.7 meters from a boundary which becomes some what absorptive above 400Hz. Note the peaks are at multiples of 100Hz since sound travels 342m/sec and the distance from the boundary is exactly 1/200th of that. The dips are very strong because two perfectly opposite signals can cancel completely (rare in practice), however the gain of perfect identical summation is only +6dB.
At very low frequencies, where the distance from the source to the boundary is less than half a wavelength, the radiated sound from the source will be directed forwards, this is called boundary coupling, and the result is an increase in output. A maximum of +6dB is eventually reached as frequency becomes lower, with +3dB being the half wavelength frequency. Putting a speaker near a boundary thus boosts bass response below the frequency where half a wavelength is enough to reach the boundary from the speaker. We can reproduce the above simulation now moving the speaker closer to the wall, 34cm. We see +3dB at 100Hz, rising to +6dB below.
Finally we look at two sound sources spaced from each other, creating an artefact called lobbing. Essentially the effect is to throw the sound in some directions more than others, and this also increases the output in that direction since more of the sound energy is focused here. We can view this as a half-space polar plot, showing the angle and level of the projected sound as though we are looking down from on top of a speaker.
As a simple example of why this happens; Imagine two sources radiating the same sound. As a listener we stand exactly between the speakers and the distance from us to each speaker is the same. So the sound from each speaker arrives at our location together and we get a good +6dB increase in level over having just one speaker.
However, we now move 1m to the left. The left speaker is now closer than the right, and the sound from this side arrives first. The waves from each speaker are now reaching our position at different points in their cycles. They might still combine nicely, or they might cancel each other entirely, or anything between those two extremes. Dependant on the distance of the listener from each sound source, and the frequency they are producing, the pattern of hot and cold areas will change. Once we get to a high enough frequency the pattern develops enough lobes that the power can be considered evenky spread out.